Embedded solitons: solitary waves in resonance with the linear spectrum

Abstract

It is commonly held that a necessary condition for the existence of solitons in nonlinear-wave systems is that the soliton's frequency (spatial or temporal) must not fall into the continuous spectrum of radiation modes. However, this is not always true. We present a new class of codimension-one solitons (i.e., those existing at isolated frequency values) that are embedded into the continuous spectrum. This is possible if the spectrum of the linearized system has (at least) two branches, one corresponding to exponentially localized solutions, and the other to radiation modes. An embedded soliton (ES) is obtained when the latter component exactly vanishes in the solitary-wave's tail. The paper contains both a survey of recent results obtained by the authors and some new results, the aim being to draw together several different mechanism underlying the existence of ESs. We also consider the distinctive property of semi-stability of ES, and moving ESs. Results are presented for four different physical models, including an extended 5th-order KdV equation describing surface waves in inviscid fluids, and three models from nonlinear optics. One of them pertains to a resonant Bragg grating in an optical fiber with a cubic nonlinearity, while two others describe second-harmonic generation (SHG) in the temporal or spatial domain (i.e., respectively, propagating pulses in nonlinear optical fibers, or stationary patterns in nonlinear planar waveguides). Special attention is paid to the SHG model in the temporal domain for a case of competing quadratic and cubic nonlinearities. An essential new result is that ES is, virtually, fully stable in the latter model in the case when both harmonics have anomalous dispersion.

When gap solitons become embedded solitons: a generic unfolding

“…Embedded soliton solution of (10) have been found for δ > 0 and γ 1,2 < 0 in [34], and for δ < 0, γ 1,2 > 0 in [11]. We shall follow the latter paper here and search for parameter values δ < 0 where our theory applies.…”

“…In [11] curves (in the (δ, q)-parameter plane) of R-symmetric embedded soliton solutions were found for k = 0.3. In a similar manner we find a curve of such solutions for k = 1.…”

“…Specifically, they occur in a two-component model when the dispersion relation has only one branch. In contrast to gap solitons, embedded solitons are not structurally stable in that they exist as codimension-one solutions, i.e., at discrete values of the frequency [34,11]. In these parameter ranges quasi-solitons (or "generalized solitary waves" [1]) in the form of solitary waves with non-decaying periodic tails are generic.…”

“…This paper concerns only the single humped variety, since to date all non-fundamental embedded solitons have been found to be (exponentially) unstable as solutions of the underlying PDEs, see e.g. [11,35].…”

Accumulation of embedded solitons in systems with quadratic nonlinearity

“…The present work was motivated by results in the two latter references, which had produced embedded solitons (ESs) in the system. ESs are distinguished by the feature that their internal frequency (spatial frequency or propagation constant, in the present context) is in resonance with radiation waves, [5]. In general, one expects to find quasi-solitons with non-vanishing oscillating tails in such a case.…”